{"paper":{"title":"Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xinchen Hu, Yutao ma","submitted_at":"2025-06-05T02:28:39Z","abstract_excerpt":"Let $X$ be a real $(\\beta=1)$ or complex $(\\beta=2)$ Ginibre ensemble. Let $\\{\\sigma_i\\}_{1\\le i\\le n}$ be the eigenvalues of $X,$ and $Z_n$ be some rescaled version of $\\max_i \\Re \\sigma_i.$ It was proved that $Z_n$ converges weakly to the Gumbel distribution $\\Lambda_{\\beta}$ with distribution function $e^{-\\frac{\\beta}{2}e^{-x}}.$ We further prove that\n  $$\\sup_{x\\in \\mathbb{R}}|\\mathbb{P}(Z_n \\leq x)-e^{-\\frac{\\beta}{2}e^{-x}}|=\\frac{25\\log \\log n}{4e \\log n}(1+o(1))$$\n  and\n  $$ W_1\\left(\\mathcal{L}(Z_n), \\Lambda_{\\beta}\\right)=\\frac{25\\log \\log n}{4\\log n}(1+o(1))$$ for sufficiently larg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.04560","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.04560/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}